to the French horn, enables the player to produce a practically perfect diatonic scale in the tonic, dominant, and subdominant keys, with the advantage of having only two valve slides to tune when changing the crook, the alteration theoretically required in the third or comma valve" being so small as to be inappreciable.

In his second, which he terms the "teleophonic" instrument, he retains the original slide, thus keeping the power of adjusting each note to accurate intonation; but he adds a single valve tuned in unison with the open D, or harmonic ninth-in other words, lowering the pitch a minor tone. This valve is worked by the forefinger of the left hand, the instrument being held exactly in the usual manner. By the use of this single valve and the slide, separately or together, it is possible to produce a complete scale, major or minor, with a perfection of intonation limited only by the skill of the player. The valve not only supplies those notes which are false or entirely wanting on the ordinary slide trumpet, including the low A flat and E flat on the higher crooks, but it greatly facilitates transposition and rapid passages.

CHAPTER VIII.

SPECIAL APPLICATIONS TO MUSIC-THE EAR AND VOICE.

Special Applications to Music.-Hitherto Sound has been considered principally in its physical aspect, with only casual reference to its musical application. But a work like the present, though precluded from entering into much detail, would be incomplete without some account of the appliances, vocal and instrumental, out of which the ancient art of music has been constructed. Even in the strictest sense, these may be regarded as apparatus; and their fabrication has in many instances preceded and cleared the way for scientific examination of their mode of operation.

Helmholtz, in drawing distinctions between the physical and æsthetical branches of acoustics, begins by pointing out as a fundamental proposition that "The system of scales, modes, and harmonic tissues, does not rest solely upon unalterable natural laws, but is at least partly also the result of æsthetical principles, which have already changed, and will still further change, with the progressive development of humanity." This, however, does not prevent their being brought under some general law. He divides all musical history into three periods :

1. The homophonic, or unison music of the ancients, still retained by Orientals and Asiatics.

2. The polyphonic music of the Middle Ages.

3. Harmonic, or modern music, dating from the sixteenth century.

The first, in any extended form, he shows to be only possible in connexion with poetry, such as the hymns and tragedies of Greece. Even in ordinary conversation, however, the voice goes through certain cadences, which are a form of

M

continuous melodic recitation, bearing specific significance in particular forms and phrases. Familiar examples are the falling inflection of the voice at the end of a phrase, and the rising cadence of interrogation.

Polyphonic music appeared in the form of discant, in which different voices, each proceeding independently and singing its own melody, had to be united in such a way as to produce either no dissonance, or only transient dissonances, which were readily resolved. In this way canonic imitation arose, as early as the twelfth century. The old ecclesiastical modes were retained, and in 1547 Glareanus distinguished twelve of them, six authentic, and six plagal, assigning to them, somewhat incorrectly, the Greek names, Ionic, Doric, Phrygian, Lydian, Mixolydian, and Æolic, by which they have been known

since.

Harmonic or modern music is marked by the independence of its construction, and the artistic connection of its parts; with this system it has become possible to compose works of greater extent, and more energetic expression than its predecessors. Its essential law is that "the whole mass of tones, and the connection of harmonics must stand in a close and perceptible relationship to some arbitrarily selected tonic; and that the mass of tone which forms the whole composition must be developed from this tonic, and must finally return to it."

Notation of Musical Tones.-Several systems of notation have been proposed with a view to distinguish the different octaves from one another. That adopted by Helmholtz, and commonly used in Germany, begins with the great or 8-foot octave, the C of which would be given by a pipe of this size, or by the lowest string of the violoncello. This is marked by capital letters as follows:

The next octave above this is termed the small or unaccented octave of 4 feet pitch distinguished by small letters.

The 2-foot octave has small letters marked with a single accent thus, c' d'e' f'g' a' b' and is termed the once-accented octave. The 1-foot octave has two of these accents c" and is termed the twice-accented octave.

Below the great octave is the 16-foot or contra-octave, distinguished by capital letters with an inverted accent below them, thus C, D, E, F, G, A, B, and below this again the still deeper, or 32-foot octave, which is marked by capitals with two inverted accents C, D, E, F, G, A,

B

"are

Mr. Ellis suggests another and very convenient notation founded on the powers of 2. "All musicians," he says, familiar with the octave, and accustomed to divide the whole range of musical sounds thus. This amounts to selecting a series of tones on the principle of continually multiplying the corresponding number of vibrations by 2. Arithmetically we are therefore bound to begin with less than 2 vibrations in a second, which, not being multiplied by two at all, may be said to commence the zero octave, and the simplest such number that can be selected is 1 itself. Then from 2 up to 4 we multiply the former number of vibrations once by 2 and have the first octave. At 4 or twice 2 we multiply twice by 2, and have the second octave. The octaves will correspond with the number of vibrations with which they begin, thus :

Oct. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Vib. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096.

The number of the octave is the index of the power of 2, giving the number of vibrations with which the octave begins. Thus 256 28 begins the 8th octave.

Supposing the notes C, D, E to occur in any octave, the number of the octave is prefixed. Thus 8E means E in the 8th octave. The difficulties arising from using different standards of pitch and temperaments are met by using some fraction as 1 in place of 1 for the initial number of vibrations. But what is commonly called the theoretical pitch, 80 = 256 vib., is the only one suggested by pure arithmetic. This simple contrivance for marking the octave obviates a vast number of difficulties.

=

40 = 16 vib. means, C in the sub-contra octave, or double under-accented or underlined great octave C or C, or twice indexed great octave C, or twice negatively indexed great octave C-2, or C-2, or 32-foot octave, or CCC octave.

5C

=

32 vib. means C in the contra octave, or an octave above all those named, or CC.

60 =

64 vib. means C in the 8-foot octave, the lowest note of the violoncello.

7C =

128 vib. means 4-foot C or small c, the lowest note of the viola or tenor violin.

=

80 256 vib. means C in the once accented or 2-foot octave, or c', or c overlined, or cc with two small c's commonly called "middle C," being on the ledger line between the bass and treble staves.

Varieties of Organ pipes.-Pipes are employed in the organ which aptly illustrate many of the principles previously mentioned. As regards material, they are either of metal or of wood; the former being composed of tin, with a greater or less admixture of lead, sometimes when very large they are made of zinc; the latter of pine wood, cedar or mahogany. In both cases they are divided into open and stopped, as before described, with an intermediate form termed halfstopped; the stopped pipe speaking an octave lower than an open pipe of the same length. But the most important classification is into flue or flute pipes, reeds, and mixtures. The two first have already been fully adverted to; the mixture stops, instead of a single pipe sounding to each note, possess several, from two or three, up to a much larger number, such as twelve or even fourteen. These are arranged in tiers upon the same supply of wind, and thus all sound together. They are tuned to the individual tones of the harmonic series given above, or to some of their octaves, such as the third, fifth, and eighth of the foundation-tone. The sesquialtera, one of the most usual mixtures, was originally a stop of two ranks only, composed of the twelfth and seventeenth intervals above the ground-tone, thus giving prominence to the third and fourth upper partials. The sesquialtera is now often made with three up to six ranks of open metal pipes. The mixture proper is more shrill and acute, comprising the seventeenth, nineteenth, twenty-second, twenty-sixth, and twenty-ninth. As in the treble, the pipes become very small, keen, and prominent, the smaller ranks are discontinued about middle C, and larger pipes, sounding an octave lower, are substituted. This alteration is called a break, and takes place also in the sesquialtera. The object of such stops is obviously to increase the brilliancy of effect by reinforcing the upper partial tones which Helmholtz has of late years shown to be always present in sound of a melodious quantity. It is very remarkable that the fact itself had been long ago discovered as a